How to intuitively understand Lagrangian interpolation?

Lagrangian interpolation is a polynomial interpolation method named after the French 18th-century mathematician Joseph Lagrange. In many practical problems, functions are used to represent some kind of internal relationship or law, and many functions can only be understood through experiments and observations. For example, if a physical quantity in practice is observed and the corresponding observations are obtained in several different places, the Lagrangian interpolation method can find a polynomial, which happens to obtain the observed values at each observation point.

Such a polynomial is called a Lagrangian (interpolated) polynomial. Mathematically, Lagrangian interpolation gives a polynomial function that passes exactly through several known points on a two-dimensional plane. Lagrangian interpolation was first discovered by the British mathematician Edward Warring in 1779 [1] and again soon after (1783) by Leonhard Euler.

In 1795, Lagrange published this interpolation method in his book "Basic Course of Mathematics in Normal Schools", and since then his name has been associated with this methodLagrange interpolation is a polynomial interpolation method. It is to construct a smooth curve using a polynomial of the least number of times, so that the curve passes through all known points. For example, the coordinates of the following 3 points are known:

x1,y1),(x2,y2),(x3,y3).Then the result is: y=y1l1+y2l2+y3l3,l1=(x-x2)(x-x3) ((x1-x2)(x1-x3)),l2=(x-x1)(x-x3) ((x2-x1)(x2-x3)),l3=(x-x1)(x-x2) ((x3-x1)(x3-x2))

Piecewise linear lagrange interpolation % command format: y=lagrange1(x0,y0,x) %x0 is the node vector, y0 is the corresponding function value vector, %x is the interpolation point vector, and the return value y is the function approximation vector at x. The interpolation method is:

The function f (x) is used to insert the function values of several points in a certain interval to make an appropriate specific function, take known values at these points, and use the value of this specific function as an approximation of the function f (x) at other points in the interval, which is called interpolation. The purpose is to estimate the value of the function at other points. Lagrangian interpolation, on the other hand, is a type of interpolation.

What do you want to say ...... for?In finance, interpolation is used to calculate the internal rate of return (IRR). <>

The results of lagrange interpolation and Newton interpolation are the same as the remainder, because they are all interpolated by n polynomials, of course, the same, the difference: Lagrange interpolation method is obtained by constructing n n + 1 n fundamental polynomials, and then linearly combining (the result is of course n times polynomial) and Newton interpolation is obtained by recursively obtaining an F(x)=F(X0)+(X-X0)f[X0,X1]+( x-x0)(x-x1)f[x0,x1,x2]..x-x0)..

x-x(n-1))f[x0,x1...xn] can be obtained by substituting it (in fact, the first floor summarizes it very deeply, sorry I haven't reached that state yet, hehe) Also, the lagrange interpolation method uses all those nodes when finding each basic polynomial, so if you need to add one more node, you need to find the basic polynomial again, and this requires a lot of engineering, so mathematicians invented the newton method, you see the formula above, If you add another node, you just need to add one (x-x0) (x-x1) after it...x-x(n-1))(x-xn)f[x0,x1...

xn,x(n+1)]? <>

A basis function is a fixed form of a function, that is, a function that only changes on the basis of this function and does not lose it. Example given n+1 control vertices pi(i=0 n), then the bezier curve is defined as: p(t)= bi,n(t)pi u [0,1] where:

bi,n(t) is called the basis function. The Lagrangian interpolation formula refers to an interpolation polynomial in which the basis function of the node is given on the node, and then the linear combination of the basis function is made, and the combination coefficient is the value of the node function. Linear interpolation is also called two-point interpolation, the value of the function y = f (x) on a given mutual difference x0, x1 is y0 = f (x0), y1 = f (x1) linear interpolation is to construct a primary polynomial p1(x) = ax + b so that it satisfies the condition p1 (x0) = y0 p1 (x1) = y1 and its geometric interpretation is a straight line through the known points a (x0, y0), b(x1, y1).

Linear interpolation is convenient to calculate and widely used, but because it uses straight lines instead of curves, it generally requires that [x0, x1] is relatively small, and f(x) changes relatively smoothly on [x0, x1], otherwise the error of linear interpolation may be very large. In order to overcome this shortcoming, simple curves are sometimes used to approximate complex curves, and the simplest curves are quadratic curves, and quadratic curves are used to approximate complex curves. To put it simply, it is to replace the original function with some functions that are easy to calculate and process.

The aim is, of course, to find an intermediate value that cannot be determined exactly, but in order to reduce error, effort, and complexity, these functions are usually replaced and combined with a primary curve (straight line) or a quadratic curve. In this way, a certain degree of accuracy can be obtained, and a balance between precision and convenience can be achieved, in a word: good and economical.

First, the meaning is different:

Both of them are given n+1 different interpolation nodes to find an n-order algebraic curve that approximates the function curve to be interpolated, which is called algebraic interpolation. Lagrange interpolation algebra and Newton interpolation both fall under the category of algebraic interpolation.

The results of lagrange interpolation and Newton interpolation are consistent with the remainder, because they are both interpolated by nth degree polynomials.

Second, the calculation is different:

The lagrange interpolation method is obtained by constructing n+1 n-order fundamental polynomials in linear combinations. The Newton method interpolation is a formula such as f(x)=f(x0)+(x-x0)f x0,x1 +(x-x0)(x-x1)f x0,x1,x2 +(x-x0)(x-x(n-1))f x0,x1,xn by recursively obtaining the quotient of each order difference.

Newtonian interpolation is characterized by:

Each additional point will not cause the previous recalculation, only the new point will be calculated.

Suppose that the values of n+1n+1 points relative to the polynomial function ff are: (x0,f(x0)),x1,f(x1)),x2,f(x2)),xn,f(xn)), find this polynomial function f.

Let's start with finding the function f1(x) that satisfies two points (x0, f(x0)), x1, f(x1)).

Suppose f1(x)=f(x0)+b1(x x0)f1(x)=f(x0)+b1(x x0), add a point, (x0, f(x0)), x1, f(x1)), x2, f(x2)), and find the function f2(x): that satisfies these three points

Suppose f2(x) = f1(x) + b2(x x0)(x x1).

Refer to the above content: Encyclopedia-Newtonian interpolation.

First, the nature is different.

1. Newtonian interpolation: a form of algebraic interpolation method. Newton's difference introduces the concept of difference quotient, making it easy to calculate when the difference node increases.

2. Lagrangian interpolation: polynomials that meet the interpolation conditions and whose order does not exceed n exist and are unique.

Second, the meaning of the formula is different.

1. Newton interpolation: As a commonly used numerical fitting method, Newton difference has been widely used in experimental analysis due to its simple calculation, many calculation points, clear logic and convenient programming.

Especially in experiments, when only discrete data points can be measured or the corresponding relationships can be represented by numerical solutions, the discrete points can be fitted with Newtonian interpolation formulas to obtain more accurate analytic values of the function.

2. Lagrangian interpolation: In many practical problems, functions are used to represent certain internal relations or laws, and many functions can only be understood through experiments and observations. If a physical quantity is actually observed and the corresponding observations are obtained at several different locations, the Lagrangian interpolation method can find a polynomial that can accurately extract the observations for each observation point.

If the nth order polynomial lj(x)(j=0,1,..n) In n+1 nodes x0 is equal to j), j,k=0,1,..n

Then the n+1 nth order polynomial l0(x), l1(x), .,ln(x) is the ,.. of nodes x0 and x1N Lagrangian interpolation basis function on xn.

For li(x)(i=0,1,..n), there is (xi's kth power) (li(x)), i from 0 to n's and =x's kth power, k=0,1,..n, especially when k = 0 there is li(x), i from 0 to n and = 1

The Lagrangian interpolation formula is derived: by the given n+1 points m1(x1,y1),m2(x2,y2) ,... on the planemn+1（xn+1，yn+1）。

Lagrange interpolation formula refers to an interpolation polynomial in which the basis function of the node is given on the node, and then the basis function is made in a linear combination, and the combination coefficient is the value of the node function. Linear interpolation is also called two-point interpolation.

Knowing that the value of the function y=f(x) on a given mutual difference x0,x1 is y0=f(x0), and the linear interpolation of y1=f(x1) is to construct a primary polynomial: p1(x)=ax+b, so that it satisfies the conditions: p1(x0)=y0, p1(x1)=y1, and its geometric interpretation is a straight line, through the known points a(x0, y0), b(x1, y1).

Linear interpolation is convenient to calculate and widely used, but because it uses straight lines instead of curves, it is generally required that [x0,x1] is relatively small, and f(x) changes relatively smoothly on [x0,x1], otherwise the error of linear interpolation with lead value may be very large. In order to overcome this shortcoming, simple curves are sometimes used to approximate complex curves.

The simplest curve is a quadratic curve, a situation where a quadratic curve is used to approximate a complex curve. In particular, for example, for two values of an independent variable, the (n=1) corresponding value of the linear function is given, and the linear function is determined. Geometrically, a straight line is determined by its two points.

Similarities and differences between Lagrangian interpolation and Newtonian interpolation:

1. Different meanings: both of them are given n+1 different interpolation nodes to find an n-order algebraic curve that approximates the function curve to be interpolated, which is called algebraic interpolation; Lagrange interpolation algebra and Newton interpolation both fall under the category of algebraic interpolation. The results of lagrange interpolation and Newton interpolation are consistent with the remainder, because they are both good values that are interpolated by nth degree polynomials.

2. Different calculations: The lagrange interpolation method is obtained by constructing n+1 nth order basic polynomials and linear combinations. The Newton method interpolation is a formula such as f(x)=f(x0)+(x-x0)f x0,x1 +(x-x0)(x-x1)f x0,x1,x2 +(x-x0)(x-x(n-1))f x0,x1,xn by recursively obtaining the quotient of each order difference.

For the function y=f(x), the value of the function on n+1 disjunctions is required to be no more than one degreenpolynomials.

such that, on the node there is.

This is called an interpolated polynomial.

Obviouslyn+1coefficients are satisfied.

The coefficient matrix of the equation is a

It is obviously a Van der Mon determinant, and it only needs to be different from each other for the system of equations to have a solution.

There is also a truncation error to consider.

Lagrangian interpolation polynomial.

First, construct a basis function.

And this function satisfies the conditions.

So the Lagrangian interpolation method is obtained.

When using the roller theorem, it is deduced that for any x belongs to the remainder of the [a, b] interpolated polynomial.

As inscribed: by Lagrangian interpolation.